在這篇論文中，我們考慮一維有震盪係數的純量傳遞方程。由於速度場的快速震盪，用數值方法解問題時我們必需使用很細的網格（ϵ 的尺度）才能得到合理的解，但這樣計算的成本會太高。我們提出一種多尺度方法，利用微觀尺度的資訊得到等效的係數。我們提供線性時的誤差估計，並進行一些數值計算。計算的範例包含單一尺度與兩個尺度形式的線性問題以及Burgers 方程與Buckley–Leverett 方程的非線性通量。我們的分析與數值結果顯示我們的方法是收斂的。此外，我們也準確地描繪非線性問題裡震波的位置。

In this thesis, we consider one-dimensional scalar transport equations with oscillatory coefficients. Due to highly oscillatory velocity field, to solve the equation numerically, we must use very fine grid size (order of ϵ) to get reasonable solutions. However, the computational cost can be very heavy. We propose a multiscale method and derive the effected coefficients from microscale equation. We give the error estimate for linear cases and present some numerical experiments. Examples include one- and two-scaled linear problems and the nonlinear flux of Burgers’ equation and Buckley–Leverett equation. Our analysis and numerical experiments show the convergence of our method. Furthermore, the position of shock wave is captured correctly for nonlinear problems.

摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction 1
2 Convergence for linear problems 2
2.1 The solution of linear problems . . . . . . . . . . . . . 2
2.2 Some convergent results . . . . . . . . . . . . . . . . . 3
3 Homogenization for nonlinear problems 6
4 Numerical method 8
4.1 Derivation of the method . . . . . . . . . . . . . . . . 8
4.2 The methodology . . . . . . . . . . . . . . . . . . . . . 9
5 Numerical analysis 9
6 Numerical result 13
6.1 Example of linear case . . . . . . . . . . . . . . . . . 13
6.2 Example of nonlinear case . . . . . . . . . . . . . . . . 15
7 Conclusion 15
Reference 18
A Solution with discontinuous velocity field 19

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